Robots need to plan their movements carefully. Path planning maps out the route, while trajectory generation adds timing to create a smooth, feasible motion. This process involves interpolating between waypoints, smoothing the path, and considering the robot's physical limitations.

Various techniques help create optimal trajectories. These include polynomial interpolation, spline fitting, and optimization methods. The goal is to balance smoothness, speed, energy efficiency, and safety while accounting for the robot's capabilities and the specific application requirements.

Path Planning and Trajectory Generation

Process of trajectory generation

• Path planning determines geometric route while trajectory generation creates time-parameterized path

• Steps in trajectory generation:

  1. Start with planned path (waypoints)
  2. Interpolate between waypoints
  3. Apply smoothing techniques
  4. Consider kinematic and dynamic constraints

• Feasibility considerations account for robot's physical limitations (velocity and acceleration bounds) and obstacle avoidance

• Smoothness criteria ensure continuity in position, velocity, and acceleration while minimizing jerk

• Iterative refinement adjusts trajectory based on constraints and optimizes for desired properties

Techniques for trajectory interpolation

• Polynomial interpolation uses cubic polynomials for smooth transitions and quintic polynomials for continuous acceleration
• Spline fitting employs B-splines for local control and smoothness and NURBS for complex curves
• Bézier curves utilize control points for shaping trajectories and De Casteljau's algorithm for evaluation
• Hermite interpolation matches position and velocity at waypoints
• Time parameterization assigns time values to waypoints ensuring consistent velocity profiles
• Constraint satisfaction incorporates kinematic limits into interpolation and adjusts coefficients to meet dynamic constraints

Trajectory Optimization and Analysis

Methods of trajectory optimization

• Objective functions minimize total time, reduce energy consumption, or maximize smoothness
• Optimization techniques include gradient descent for local optimization, genetic algorithms for global search, and Model Predictive Control for real-time optimization
• Convex optimization formulates trajectory generation as a convex problem using tools (CVX, CVXOPT)
• Dynamic Programming breaks down trajectory into subproblems using Bellman equation for optimal substructure
• Quadratic Programming minimizes quadratic cost functions subject to linear constraints
• Constrained optimization incorporates obstacle avoidance and respects joint limits and actuator capabilities

Trade-offs in trajectory planning

• Smoothness vs time optimality: smoother trajectories may take longer to execute while time-optimal trajectories may have abrupt changes
• Feasibility vs optimality: optimal trajectories may push robot limits while feasible trajectories ensure safe execution
• Energy efficiency vs speed: slower movements often consume less energy while rapid movements may require more power
• Computation time vs trajectory quality: real-time constraints may limit optimization while offline planning allows for more refined trajectories
• Robustness to uncertainties: smoother trajectories often more robust to disturbances while optimal trajectories may be sensitive to model inaccuracies
• Application-specific considerations: manufacturing balances precision vs throughput, mobile robots consider battery life vs task completion time, manipulation weighs force control vs speed of operation
• Safety and human interaction: collaborative robots use slower, smoother trajectories while isolated industrial settings allow rapid movements